In a large pot, bring several inches of water to a boil for cooking the bitterness out of the rapini. Bring a larger pot of water to a boil for the pasta. Salt the boiling water for the rapini, add the rapini and cook 2 to 3 minutes; using a slotted spoon, transfer to a bowl of ice water. Let cool briefly; drain. Chop into bite-size pieces.
In a large, deep skillet, heat the oil, two turns of the pan, over medium-high. Add the sausage and cook, breaking up with a spoon, until browned, about 5 minutes. Add the chile and garlic and stir until fragrant, 1 to 2 minutes. Add the rapini, the thyme and nutmeg to taste; stir until heated through, about 3 minutes. Add the tomato paste and stir until fragrant, 1 to 2 minutes. Add the vermouth and cook, stirring often, about 2 minutes. Add the stock and reduce the heat to low.
Meanwhile, salt the boiling water for the pasta. Add the pasta; cook to al dente.
Drain the pasta, add it to the skillet and toss to coat. Divide the pasta among bowls and top with the ricotta salata.